Question

Consider an LRC circuit with \(L=1.00 \mathrm{H}, R=1.00 \times 10^{2} \Omega, C=\) \(1.00 \times 10^{-4} \mathrm{~F}\), and \(V=1.00 \times 10^{3} \mathrm{~V}\). Suppose that no charge is present and no current is flowing at time \(t=0\) when a battery of voltage \(V\) is inserted. Find the current and the charge on the capacitor as functions of time. Describe how the system behaves over time.

Short answer

The charge on the capacitor as a function of time is given by \(q(t) = CV(1 - e^{-t/RC})\) and the current through the circuit as a function of time is \(i(t) = (V/R)e^{-t/RC}\). When the system is first switched on, the capacitor starts to charge from zero and the current decreases from \(V/R\). Over time, the capacitor charge increases exponentially to a maximum value \(Qmax= CV\), while the current decays exponentially to zero.

Step-by-step solution

Find the initial conditions

For \(t=0\), the charge \(q(0)=0\) (as it is specified that there is no charge present initially), and the current \(i(0)=V/R\) (as there is no inductor to prevent the flow of current when the circuit is first turned on).

Write differential equation for the circuit

For the LRC circuit, the law of Kirchhoff gives \(-L \cdot di(t)/dt-R \cdot i(t)-q(t)/C=0\). Multiply the zero by \(C\) then differentiate between the time to get \((L \cdot C) \cdot d²q(t) / dt² + (R \cdot C) \cdot dq(t)/dt+q(t)=0\).

Find the charge over time

The charge over time \(q(t)\) is given by: \(q(t) = Qmax \cdot (1 - e^{-t/\tau})\) where the maximum charge \(Qmax = CV\) and the time constant \(\tau = RC\).

Find the current over time

When looking at the current, \(i(t) = dq(t)/dt = (V/R)e^{-t/\tau}\) where \(R\) is the resistance, \(V\) is the voltage, and \(t\) is time.

Describe the system behavior over time

Through these expressions, we see that the charge on the capacitor and the current through the circuit start at 0 and \(V/R\) respectively when the circuit is initially switched on. As time progresses, the capacitor gradually charges up (with an exponential increase in charge) to a maximum charge \(Qmax\), and the current gradually declines (with an exponential decay) to zero.

Key concepts

Differential Equations in Physics

Differential equations play a pivotal role in physics as they describe the continuous changes in physical systems. They are mathematical representations of the rates at which these changes occur. In an LRC circuit, which is an electrical circuit composed of an inductor (L), a resistor (R), and a capacitor (C), the behavior over time is often governed by second-order differential equations.

For instance, Kirchhoff's law leads to the formation of a differential equation that balances the voltage drops across the elements in the circuit. Specifically, the changing current through the inductor creates a back-emf that opposes the change in current, encapsulated in the term \(L \frac{di}{dt}\). The voltage drop across the resistor is proportional to the current, given as \(Ri\). The voltage across the capacitor relates to the charge present, formulated as \(\frac{q}{C}\). Combining these elements in accordance with Kirchhoff's voltage law, we end up with the differential equation \(L \frac{d^{2}q}{dt^{2}} + R \frac{dq}{dt} + q = 0\).

This equation helps to determine the dynamic response of an LRC circuit to changes over time, including how the charge and current evolve after a battery is connected.

Time Constant for Circuit

The time constant is a crucial value in circuit analysis as it quantifies how quickly the voltage and current in an electrical circuit responds to changes over time. Specifically, it is a measure of the time it takes for a system to change significantly from its previous state, often described as the time taken for a response to reach approximately 63.2% of its final value in a first-order linear system.

The time constant \( \tau \) in an LRC circuit without a driving voltage is calculated as the product of resistance and capacitance, \( \tau = RC \). It effectively combines the effects of the resistor's ability to impede current and the capacitor's ability to store charge to predict the rate at which the circuit will reach a steady state. A larger time constant means that the system reacts more sluggishly to changes; conversely, a smaller time constant corresponds to a faster reacting system.

Understanding the time constant for a given circuit is paramount when analyzing transient responses, such as in the charging and discharging processes of a capacitor within an LRC circuit.

Exponential Charge and Discharge

The exponential nature of charging and discharging in an LRC circuit can be a bit intimidating at first, but it is, in fact, an elegant manifestation of how electrical energy is stored and released over time. When a battery is connected to an LRC circuit with no initial charge or current, the capacitor begins to store charge, and this process can be described by an exponential function.

During the charging phase, the charge on the capacitor \(q(t)\) can be represented by \(q(t) = Q_{max}(1 - e^{-t/\tau})\), where \(Q_{max}\) is the maximum charge the capacitor can hold, \(e\) is the base of the natural logarithm, and \(\tau\) is the time constant. This formula reflects that the charge starts at zero and asymptotically approaches \(Q_{max}\) as time progresses – a process characterized by a sharp increase initially that slows down as it gets closer to the maximum charge.

Similarly, the current in the circuit decreases exponentially over time, with \(i(t) = (V/R)e^{-t/\tau}\). The current is initially at its peak when there is no opposing EMF from the inductor, then gradually decreases to zero as the circuit reaches equilibrium. This relationship underscores the concept that energy is not lost but is transferred between the inductor, capacitor, and resistor, with each influencing how quickly or slowly the current changes.